Few sites used Login with LinkedIn simply because the API is actually pretty difficult. Additionally, LinkedIn is seen as more niche-y because of it's business focus vs Facebook which is pretty much for everyone.
As for issues, they do happen. However, LinkedIn is much, much more stable than Facebook has been. So far this year, this is the 2nd change that has caused functionality to break. The first, in June broke logging in altogether, but was fixed by LinkedIn within about 20 hours of it being reported. This recent change is new, but is an issue with JLinked specifically, so we'll need to fix this one.
Facebook, has had more than 4 'breaks' since the beginning of this year. The majority have been, again, on Facebook's end to fix and not problems with our extensions. When things need to be changed with us, we are usually very quick, depending on the severity. For avatars, while it is important, we don't rate the issue as critical since overall things are still functional (no errors; a feature simply doesn't work as expected).
Hope that makes sense. I may try tonight or tomorrow morning to see if there's a way we can implement the code on your site in a quick, get-it-done fashion before the full release.
Thanks,
Alex